Difference between revisions of "1961 IMO Problems/Problem 2"
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+ | Let <math>a</math>, <math>b</math>, and <math>c</math> be the lengths of a triangle whose area is ''S''. Prove that | ||
<math>a^2 + b^2 + c^2 \ge 4S\sqrt{3}</math> | <math>a^2 + b^2 + c^2 \ge 4S\sqrt{3}</math> | ||
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In what case does equality hold? | In what case does equality hold? | ||
− | {{ | + | ==Solution== |
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+ | Substitute <math>S=\sqrt{s(s-a)(s-b)(s-c)}</math>, where <math>s=\frac{a+b+c}{2}</math> | ||
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+ | This shows that the inequality is equivalent to <math>a^2b^2+b^2c^2+c^2a^2\le a^4+b^4+c^4</math>. | ||
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+ | This can be proven because <math>a^2b^2\le\frac{a^4+b^4}{2}</math>. The equality holds when <math>a=b=c</math>, or when the triangle is equilateral. | ||
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+ | {{IMO box|year=1961|num-b=1|num-a=3}} |
Revision as of 11:37, 28 December 2007
Problem
Let , , and be the lengths of a triangle whose area is S. Prove that
In what case does equality hold?
Solution
Substitute , where
This shows that the inequality is equivalent to .
This can be proven because . The equality holds when , or when the triangle is equilateral.
1961 IMO (Problems) • Resources | ||
Preceded by Problem 1 |
1 • 2 • 3 • 4 • 5 • 6 | Followed by Problem 3 |
All IMO Problems and Solutions |